Geometry of the Wiman–Edge pencil and the Wiman curve

Abstract

The Wiman–Edge pencil is the universal family Ct, t∈ B of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group A5. The curve C, discovered by Wiman in 1895 (Ueber die algebraische Curven von den Geschlecht p= 4 , 5 and 6 welche eindeutige Transformationen in sich besitzen) and called the Wiman curve, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group S5. In this paper we give an explicit uniformization of B as a non-congruence quotient Γ \ H of the hyperbolic plane H, where Γ<PSL2(Z) is a subgroup of index 18. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of Ct into 10 lines (resp. 5 conics) whose intersection graph is the Petersen graph (resp. K5). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve C itself as the quotient Λ \ H, where Λ is a principal level 5 subgroup of a certain “unit spinor norm” group of Möbius transformations. We then prove that C is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.